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I have a formula for certain coefficients in terms of Littlewood-Richardson coefficients and $p$-cores and $p$-quotients of partitions ($p$ is a prime). I would like to obtain some positivity conditions from this formula.

I am looking for results that relate Littlewood-Richardson coefficients with cores and quotients of a partition. Some examples of the type of result include:

  1. Given partition $\lambda, \mu_1,\mu_2$ with $c^{\lambda}_{\mu_1,\mu_2}>0$, what can be said about the p-cores and p-quotients of $\mu_1,\mu_2$?

  2. Given a partition $\lambda$ with $p$-quotient $(\nu_0,\dotsc,\nu_{p-1})$, for what partitions $\nu$ is $c^{\nu}_{\nu_0,\dotsc,\nu_{p-1}}$ nonzero?

In general I would like any references that connect both these concepts. I apologise if this is a vague request.

Thanks in advance.

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    $\begingroup$ The closest I can think of is the Littlewood map, and the modular hook-formula: symmetricfunctions.com/borderStripTableaux.htm#littlewoodMap I have not seen LR-coefficients from this perspective $\endgroup$
    – Per Alexandersson
    Commented Sep 16, 2021 at 7:09
  • $\begingroup$ 1. Obviously, the size of the $p$-core of $\lambda$ is congruent modulo $p$ to the sum of the sizes of the $p$-cores of $\mu_1$ and $\mu_2$. I would be somewhat surprised if anything other than this would hold for the $p$-cores alone. ... $\endgroup$
    – darij grinberg
    Commented Sep 16, 2021 at 15:17
  • $\begingroup$ ... Some Sage code to experiment with $p = 2$. (Here, I encode each $2$-core by its length.) $\endgroup$
    – darij grinberg
    Commented Sep 16, 2021 at 15:17

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